! Except for a fully filled subshell, we rarely presume to know which of the two possible spin states individual electrons have (m s = ±½).

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1 Terms of Free Ions with d n Configurations! The usual notation for electronic configurations (e.g., 3d 2 ) does not tell us which specific orbitals are occupied, except when a degenerate set of orbitals is half filled or fully filled.! Except for a fully filled subshell, we rarely presume to know which of the two possible spin states individual electrons have (m s = ±½). Example: There are ten ways of arranging the single electron in the configuration nd 1 (n = 3, 4, 5,...) m l = -2 m l = -1 m l = 0 m l = +1 m l = +2! The number of possible arrangements increases greatly with additional electrons.

2 Electron Arrangements for nd 2 (Same Spins)! Consider the 20 possible ways of arranging electrons for the configuration nd 2 (n = 3, 4, 5,...), in which the two electrons have the same spins (both m s = +½ or m s = ½) : m l = -2 m l = -1 m l = 0 m l = +1 m l = +2

3 Electron Arrangements for nd 2 (Opposite Spins)! There are 25 additional configurations with opposite spins. m l = -2 m l = -1 m l = 0 m l = +1 m l = +2 ¾ ¾ ¾ ¾ ½ ¾ ¾ ¾ ½ ½ ¾ ¾ ½ ½ ½ ¾ ½ ½ ½ ½

4 Microstates! Each set of individual m l and m s values constitutes a microstate of the configuration.! Some microstates may be allowable arrangements in the ground state, and others may be allowable arrangements in some higher-energy excited state.! With additional electrons the number of microstates rises dramatically.! For any allowed number of electrons in a set of degenerate orbitals (called equivalent electrons), the number of possible microstates is given by D t = N o = D t ' (2N o )! (2N o & N e )!N e! number of possible microstates = total degeneracy of the configuration number of degenerate orbitals in the set or subshell N e = number of electrons in the configuration Example: nd 2 D t ' (2 5)! [(2 5) & 2)]!2! ' 10! 8! 2! ' (40320)(2) ' 45

5 Microstates for nd 1 10 configuration d 1 d 2 d 3 d 4 d 5 d 6 d 7 d 8 d 9 d 10 microstates L The number of microstates (the total degeneracy) for a configuration nd x is the same as for nd 10 x.

6 Microstates and Terms! In any microstate both the individual orbital magnetic moments (related to m l ) and spin magnetic moments (related to m s ) will interact with one another, resulting in an energy state or term for the configuration.! Except for a fully filled configuration, no one microstate uniquely gives rise to a particular energy for the configuration.! A number of microstates generally contribute to a single term.! Terms are usually degenerate according to the number of microstates giving rise to them.! In a ligand field, degeneracies among all the microstates comprising a particular free-ion term may be partially lifted.

7 Russell-Saunders Coupling Scheme! The ways in which individual m l and m s values interact are not easily evaluated for a real atom or ion.! In fact, the notion that we can assign individual m l and m s values to the electrons and assess their interactions on that basis is really an extreme extension of the one-electron wave mechanical model.! The Russell Saunders coupling scheme is a reasonably good first approximation for assessing the origins of the term energies in many cases in terms of summations of individual electron m l and m s values.! The Russell-Saunders coupling scheme can be applied successfully to interpreting the term energies of first and second row transition metals " Less successful with the third row transition elements " Hopelessly inadequate with f-block transition elements (i.e., lanthanides and actinides).

8 Free-Ion Term Symbols! The various terms that can exist for a particular configuration are indicated by a term symbol of the form 2S + 1 L " L is related to the overall orbital angular momentum for the system of electrons. Values of L may be 0, 1, 2,... " S is related to the overall spin angular momentum for the system of electrons. Values of S may be 0, 1/2, 1, 3/2,... " L and S are analogous to the quantum numbers l and s for single electrons.! The left superscript number 2S + 1 is the multiplicity of the term.

9 The J Quantum Number! A third quantum number, J, is often added to term symbols as a right subscript. 2S+1 L J " J relates to the total angular momentum arising from spin-orbital coupling. " Allowed J values are L + S, L + S - 1, L + S - 2,..., *L - S*. " For a given L value, the various values of J represent closely spaced energy sublevels of the term energy. " 2S + 1 equals the number of J values (and hence the number of sublevels) for the particular L value when L > S. " When L = 0 or L < S the number of J values is 2L + 1. " For a given L value the energy differences between the sublevels of various J values are small and can be ignored for our purposes. L We will omit the J values from our term symbols.! The multiplicity indicated by the superscript 2S + 1 is retained, because it relates directly to the spin state (and hence the number of unpaired electrons) of the term. L 2S + 1 is often called the spin multiplicity and used without reference to J.

10 Relationship of L to Resultant Orbital Angular Momentum and Resultant Orbital Magnetic Moment L = Overall orbital angular momentum quantum number = Resultant orbital quantum number! L is related to the resultant orbital angular momentum and to the resultant orbital magnetic moment of the system, obtained by vectorial addition of the vectors related to the l quantum numbers of the individual electrons. " The orbital angular momentum for individual electrons has a magnitude of [l(l + 1)] ½ (h/2π) " Likewise the resultant orbital angular momentum has a magnitude of [L(L + 1)] ½ (h/2π).! There are a variety of ways in which the individual l values can add vectorially, so a variety of L values can result for a given configuration of electrons. " This is true even when all electrons have the same individual l values.

11 Vector Addition of Individual l Vectors to Give L Values for Configurations p 2 and d 2 p2 l = 1 l = 1 L = 2 (D) l = 1 l = 1 l = 1 l = 1 L = 1 (P) L = 0 (S) d 2 l = 2 l = 2 L = 4 (G) l = 2 l = 2 l = 2 L = 3 (F) l = 2 L = 2 (D) l = 2 l = 2 l = 2 l = 2 L = 1 (P) L = 0 (S)

12 Term Symbol Notation L value State S P D F G H I K...! Capital letter designations follow orbital notations (s, p, d, f)! After L = 2 notation proceeds alphabetically (with the omission of J, to avoid confusion with the total angular momentum quantum number, J). Example: Configuration p 2 gives rise to the terms S, P, and D Example: Configuration d 2 gives rise to the terms S, P, D, F, and G

13 Relationship Between L and M L! For a given term the magnitude of the resultant orbital angular momentum is fixed as [L(L + 1)] ½ (h/2π). " However, the vector for the momentum can have a number of allowed orientations in space relative to an applied magnetic field, which defines the z direction of the system. " The various allowed orientations are associated with the overall orbital magnetic quantum number, M L, which can take on the 2L + 1 values M L = L, L - 1,..., 1 - L, -L " 2L + 1 may be regarded as the orbital multiplicity or orbital degeneracy of the term. " Each orientation has a projection on z whose magnitude is M L (h/2π).

14 Example: M L Orientations for a D Term! A D term (L = 2, 2L + 1 = 5) has five possible orientations corresponding to M L = +2, +1, 0, -1, -2.! The magnitude of the resultant orbital angular momentum is [L(L + 1)] ½ (h/2π) = [(2)(2+1)] ½ (h/2π) = /6 (h/2π)! Each orientation of the resultant orbital angular momentum has a projection on z whose magnitude is M L (h/2π) = +2h/2π, +h/2π, 0, h/2π, 2 h/2π +z +2h/2π +h/2π M L = +2 M L = +1 0 M L = 0 -h/2π -2h/2π M L = -1 M L = -2 -z

15 Relationship Between M L and m l! In the Russell-Saunders coupling scheme values of M L can be obtained as the sum of the m l values of the individual electrons; i.e., M L = Gm l.! Thus it is possible to assign an M L value for each and every microstate of a configuration.! Since M L represents the possible orientations of the orbital angular momentum vector, it follows that a given L value must arise from a complete set of microstates with the 2L + 1 values M L = L, L - 1,..., 1 - L, -L, which identify these orientations.

16 Overall Spin Quantum Number, S! The overall spin quantum number, S, defines the spin state of the term, and 2S + 1 defines the spin multiplicity. S 2S + 1 State Multiplicity 0 1 singlet 1/2 2 doublet 1 3 triplet 3/2 4 quartet (quadruplet) ! S is related to the resultant spin angular momentum and to the resultant spin magnetic moment of the system.! S can be obtained by vectorial addition of the spin angular momentum vectors related to the s quantum numbers of the individual electrons.! The magnitude of the resultant spin angular momentum is [S(S + 1)] ½ (h/2π)! S is related to an overall spin magnetic quantum number, M S, whose allowed values are M S = S, S + 1,..., 1 - S, -S

17 Allowed Orientations of the Spin Angular Momentum! The 2S + 1 values indicate the allowed orientations of the spin angular momentum vector relative to an applied magnetic field, which defines the z direction of the system.! The spin multiplicity, given as 2S + 1, represents the spin degeneracy of a particular spin state.! For a given spin state the magnitude of the spin angular momentum is fixed as [S(S + 1)] ½ (h/2π), but its projections on the z axis in the allowed orientations are given by M S (h/2π). Example: S = 1, M S = -1, 0, +1 +z +h/2π M S = +1 0 M S = 0 -h/2π M S = -1 -z

18 Relationship Between M S and m s! In the Russell-Saunders scheme we assume that M S is the sum of m s values of the individual electrons; i.e., where m s = ±½. M S = Gm s! Each microstate can be assigned a value of M S.! Since M S represents the possible orientations of the spin angular momentum vector, it follows that a given S value must arise from a complete set of microstates with the 2S + 1 values M S = S, S - 1,..., 1 - S, -S.

19 Relationship Between M L and M S of Microstates and L and S of Terms L A term having particular values of both L and S must arise from the set of microstates that has the necessary 2L + 1 values of M L and also the necessary 2S + 1 values of M S.! This means that one can identify all the allowed terms of a configuration by systematically arranging all microstates in such a way as to be able to cull the sets of M L and M S values with the appropriate ranges that define the various terms.! This is a straightforward but tedious process, especially for configurations with large numbers of microstates. ( We will not concern ourselves with the mechanics of this task, except to note that a variety of techniques have been developed to carry out the labor. 1 1 Two good methods have been described by K. E. Hyde, J. Chem. Educ., 1975, 52, 87 and E. R. Tuttle, Amer. J. Phys., 1967, 35, 26.

20 Equivalent Electrons Have Fewer Possible Terms Example: d 2! We have seen by vector addition that the configuration d 2 gives rise to the terms S, P, D, F, G! The only possible spin states are S = 0 (paired) and S = 1 (unpaired). " Thus, the spin multiplicities of the terms can only be singlets and triplets. " If the two electrons are in different subshells (e.g., 3d 1 4d 1 ) all terms will occur as both singlets and triplets. " If we stipulate that the two electrons are equivalent, meaning a configuration nd 2 within the same subshell, the Pauli exclusion principle will limit the possible combinations of m l and m s, so only certain terms can be singlet or triplets. For the configuration nd 2 the allowed terms are 1 S, 3 P, 1 D, 3 F, 1 G

21 Relationship Between Total Degeneracy and Term Degeneracies Example: nd 2 Y 1 S, 3 P, 1 D, 3 F, 1 G! We know that the terms for two equivalent d electrons arise from 45 microstates. " Therefore the sum of the degeneracies of all these terms must equal this number. L The degeneracy of each term, equivalent to the number of microstates giving rise to it, is the product of its spin degeneracy times its orbital degeneracy; i.e., (2S + 1)(2L + 1).! For the set of terms for nd 2 we have Term 1 S 3 P 1 D 3 F 1 G D t Degeneracy (1)(1) = 1 (3)(3) = 9 (1)(5) = 5 (3)(7) = 21 (1)(9) = 9 45

22 Terms for Free Ion d n Configurations d n Free Ion Terms D t d 0, d 10 1 S 1 d 1, d 9 2 D 10 d 2, d 8 1 S, 1 D, 1 G, 3 P, 3 F 45 d 3, d 7 2 P, 2 D(2), 2 F, 2 G, 2 H, 4 P, 4 F 120 d 4, d 6 1 S(2), 1 D(2), 1 F, 1 G(2), 1 I, 3 P(2), 3 D, 3 F(2), 3 G, 3 H, 5 D 210 d 5 2 S, 2 P, 2 D(3), 2 F(2), 2 G(2), 2 H, 2 I, 4 P, 4 D, 4 F, 4 G, 6 S 252 L Note that in each case the total degeneracy of terms is equal to the calculated value of D t.! The ground state term can be identified by applying Hund's rules.! In general the actual energies of the terms, and hence their relative ordering, must be determined from analysis of spectroscopic data.